DEPENDENCE OF THE STRUCTURAL PROPERTIES OF THE DUAL BASIS ON THE TYPE OF THE CHARACTERISTIC POLYNOMIAL

Abstract

The relevance of the topic is connected with the solution of problems of processing of recurrent sequences. The method used to determine the initial phase of a pseudorandom sequence constructed by the characteristic polynomial allows its processing directly in the receiving process. This approach does not require mandatory accumulation of the entire sequence, which significantly increases the efficiency of the system, which uses a dual basis as a method of decoding recurrent sequences. The method of decoding using the dual basis of the Galois field is an effective means of protection against errors in a digital data transmission system, where the codes of BCH, Reed-Solomon and other varieties of recurrent sequences can be used as noise resistant codes. At the same time, the calculation of the elements of the basis dual to the left power basis generates a number of problems related to mathematical calculations over the elements of a finite field. In this paper, the methods of finding the elements of the dual basis are considered, the variants of simplified procedures for their calculation are proposed, and the task is to study the structural properties of the dual basis, which depend on the type of the characteristic polynomial. Research subject. The work is devoted to the study of binary Galois fields, in particular, the properties of the left power and dual basis are affected. Method. On the example of M-sequences, the procedure for finding the elements of the dual basis through the trace function is considered in detail using the mathematical apparatus developed by O. S. Korovitsky. Core results. The analysis of the given properties of the dual basis allowed to reveal new structural dependences between its elements. Practical relevance. The revealed structural dependences between the elements of the dual basis of the Galois field make it possible to significantly simplify the implementation of the procedure for the formation of its elements.